Optimal. Leaf size=124 \[ -\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{x}-\frac {a \sqrt {a x-1} \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {a \sqrt {a x-1} \cosh ^{-1}(a x)^2}{\sqrt {1-a x}}-\frac {2 a \sqrt {a x-1} \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )}{\sqrt {1-a x}} \]
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Rubi [A] time = 0.44, antiderivative size = 174, normalized size of antiderivative = 1.40, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5798, 5724, 5660, 3718, 2190, 2279, 2391} \[ -\frac {a \sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {(1-a x) (a x+1) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 5724
Rule 5798
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^2}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}-\frac {\left (2 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)}{x} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}-\frac {\left (2 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}-\frac {\left (4 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}-\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (2 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}-\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}-\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 111, normalized size = 0.90 \[ \frac {a \sqrt {\frac {a x-1}{a x+1}} (a x+1) \left (\text {Li}_2\left (-e^{-2 \cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x) \left (\frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)}{a x}-\cosh ^{-1}(a x)-2 \log \left (e^{-2 \cosh ^{-1}(a x)}+1\right )\right )\right )}{\sqrt {-((a x-1) (a x+1))}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )^{2}}{a^{2} x^{4} - x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 241, normalized size = 1.94 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, a x -1\right ) \mathrm {arccosh}\left (a x \right )^{2}}{x \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2} a}{a^{2} x^{2}-1}+\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}}{\sqrt {a x + 1} \sqrt {-a x + 1} x} - \int \frac {2 \, {\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}{{\left (\sqrt {a x + 1} a x^{2} + {\left (a x + 1\right )} \sqrt {a x - 1} x\right )} \sqrt {-a x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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